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Chapter 4: Problem 20

A taste test is conducted between two brands of diet cola, Brand \(\mathrm{A}\) and \(\mathrm{Brand} \mathrm{B},\) to determine if there is evidence that more people prefer Brand A. A total of 100 people participate in the taste test. (a) Define the relevant parameter(s) and state the null and alternative hypotheses. (b) Give an example of possible sample results that would provide strong evidence that more people prefer Brand A. (Give your results as number choosing Brand \(\mathrm{A}\) and number choosing Brand B.) (c) Give an example of possible sample results that would provide no evidence to support the claim that more people prefer Brand \(\mathrm{A}\). (d) Give an example of possible sample results for which the results would be inconclusive: the sample provides some evidence that Brand \(\mathrm{A}\) is preferred but the evidence is not strong.

Short Answer

Expert verified
In this example, the null hypothesis is that there is no preference between Brands A and B (50% of people like each brand). The alternative hypothesis is that more than 50% of people like Brand A. Strong support for Brand A would be shown with, for example, 80 out of 100 taste testers preferring Brand A. No support would be shown by an even split (50 for Brand A, 50 for Brand B). Inconclusive results might occur if 60 prefer Brand A and 40 prefer Brand B.

Step by step solution

01

- Define the parameter and state the null and alternative hypotheses

Parameter: \( p_A \), the proportion of all people who prefer Brand A over Brand B. \[ \] Null Hypothesis, \( H_0 \): \( p_A = 0.5 \). This means that 50% of people prefer Brand A, implying no preference between the two brands. \[ \] Alternative Hypothesis, \( H_A \): \( p_A > 0.5 \). This means that more than 50% of people prefer Brand A, implying preference for Brand A.
02

- Give example for strong evidence supporting alternative hypothesis

Assume that of the 100 people who took part in the taste test, 80 chose Brand A and 20 chose Brand B. Here, the proportion who prefer Brand A, which is 0.8 (80/100), is substantially more than 0.5, providing strong evidence in favor of the alternative hypothesis.
03

- Give example for no evidence supporting alternative hypothesis

Assume that of the 100 people who took part in the taste test, 50 chose Brand A and 50 chose Brand B. Here, the proportion who prefer Brand A and Brand B, both are 0.5 (50/100), which supports the null hypothesis and provides no evidence of preference for Brand A.
04

- Give example for inconclusive results

Assume that of the 100 people who took part in the taste test, 60 chose Brand A and 40 chose Brand B. Here, the proportion who prefer Brand A, which is 0.6, is more than 0.5 but not significantly more to draw a clear conclusion. Therefore, the results are inconclusive and provide some evidence but not strong evidence in favor of the alternative hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Taste Test
A taste test is a fun and practical way to gather feedback on different products, like sodas, ice creams, or any other consumables. It involves participants trying two or more versions of a product and choosing their favorite. This type of test helps companies understand consumer preferences, which can inform product development and marketing strategies.
For a taste test to be effective, it usually needs a good number of participants to ensure reliable results. In our example, 100 people participated, which is a fairly robust sample size. This allows for meaningful analysis of the preferences between two diet colas: Brand A and Brand B.
Understanding how people make choices in a taste test helps us set up hypotheses based on their preferences, which leads us to hypothesis testing. Analyzing results from a taste test can tell us whether consumers have a significant preference for one product over another. This understanding is crucial for businesses to make informed decisions.
Null Hypothesis
The null hypothesis is a foundational concept in statistics, especially relevant in hypothesis testing. It is a statement that there is no effect or no difference, and it forms the basis of many statistical tests.
In the context of our taste test, the null hypothesis (often denoted as \( H_0 \)) is that there is no preference between the two brands of diet cola. Mathematically, this is expressed as \( p_A = 0.5 \), meaning exactly 50% of the participants would prefer Brand A, assuming no bias or preference exists.
It's important to start with this assumption because it allows us to use statistical methods to test whether there's enough evidence to reject it. In hypothesis testing, we either reject or fail to reject the null hypothesis based on our analysis. The null hypothesis is critical because it reflects a baseline expectation, helping to evaluate changes or differences in the observed data.
Proportion Analysis
Proportion analysis helps us understand and interpret the results of categorical data, especially when looking at preferences or counts in experiments like taste tests. It involves comparing proportions, or percentages, to determine if one group is preferred over another.
In our example, we're analyzing the proportion of participants who selected Brand A over Brand B. The parameter we're interested in is \( p_A \), the proportion who prefer Brand A. The analysis revolves around comparing this proportion to 0.5, which represents no preference.
When calculating proportions, we divide the number of people who chose a particular option by the total number of participants. In a taste test:
  • Strong evidence for a preference is observed if the proportion is significantly greater than 0.5 (e.g., 80 out of 100 choose Brand A, giving a proportion of 0.8).
  • No evidence for preference happens when the proportion equals 0.5 (e.g., 50 for each brand).
  • Inconclusive evidence occurs when the proportion is slightly more than 0.5 but not enough to rule out sampling variability (e.g., 60 out of 100 choose Brand A, proportion is 0.6).
Understanding proportion analysis is essential as it informs whether we can confidently claim one product is favored, influencing evidence-based decisions.

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Most popular questions from this chapter

Give null and alternative hypotheses for a population proportion, as well as sample results. Use StatKey or other technology to generate a randomization distribution and calculate a p-value. StatKey tip: Use "Test for a Single Proportion" and then "Edit Data" to enter the sample information. Hypotheses: \(H_{0}: p=0.5\) vs \(H_{a}: p \neq 0.5\) Sample data: \(\hat{p}=28 / 40=0.70\) with \(n=40\)

Could owning a cat as a child be related to mental illness later in life? Toxoplasmosis is a disease transmitted primarily through contact with cat feces, and has recently been linked with schizophrenia and other mental illnesses. Also, people infected with Toxoplasmosis tend to like cats more and are 2.5 times more likely to get in a car accident, due to delayed reaction times. The CDC estimates that about \(22.5 \%\) of Americans are infected with Toxoplasmosis (most have no symptoms), and this prevalence can be as high as \(95 \%\) in other parts of the world. A study \(^{37}\) randomly selected 262 people registered with the National Alliance for the Mentally Ill (NAMI), almost all of whom had schizophrenia, and for each person selected, chose two people from families without mental illness who were the same age, sex, and socioeconomic status as the person selected from NAMI. Each participant was asked whether or not they owned a cat as a child. The results showed that 136 of the 262 people in the mentally ill group had owned a cat, while 220 of the 522 people in the not mentally ill group had owned a cat. (a) This is known as a case-control study, where cases are selected as people with a specific disease or trait, and controls are chosen to be people without the disease or trait being studied. Both cases and controls are then asked about some variable from their past being studied as a potential risk factor. This is particularly useful for studying rare diseases (such as schizophrenia), because the design ensures a sufficient sample size of people with the disease. Can casecontrol studies such as this be used to infer a causal relationship between the hypothesized risk factor (e.g., cat ownership) and the disease (e.g., schizophrenia)? Why or why not? (b) In case-control studies, controls are usually chosen to be similar to the cases. For example, in this study each control was chosen to be the same age, sex, and socioeconomic status as the corresponding case. Why choose controls who are similar to the cases? (c) For this study, calculate the relevant difference in proportions; proportion of cases (those with schizophrenia) who owned a cat as a child minus proportion of controls (no mental illness) who owned a cat as a child. (d) For testing the hypothesis that the proportion of cat owners is higher in the schizophrenic group than the control group, use technology to generate a randomization distribution and calculate the p-value. (e) Do you think this provides evidence that there is an association between owning a cat as a child and developing schizophrenia? \(^{38}\) Why or why not?

State the null and alternative hypotheses for the statistical test described. Testing to see if there is evidence that a mean is less than 50 .

Hypotheses for a statistical test are given, followed by several possible confidence intervals for different samples. In each case, use the confidence interval to state a conclusion of the test for that sample and give the significance level used. Hypotheses: \(H_{0}: \mu=15\) vs \(H_{a}: \mu \neq 15\) (a) \(95 \%\) confidence interval for \(\mu: \quad 13.9\) to 16.2 (b) \(95 \%\) confidence interval for \(\mu: \quad 12.7\) to 14.8 (c) \(90 \%\) confidence interval for \(\mu: \quad 13.5\) to 16.5

By some accounts, the first formal hypothesis test to use statistics involved the claim of a lady tasting tea. \({ }^{11}\) In the 1920 's Muriel Bristol- Roach, a British biological scientist, was at a tea party where she claimed to be able to tell whether milk was poured into a cup before or after the tea. R.A. Fisher, an eminent statistician, was also attending the party. As a natural skeptic, Fisher assumed that Muriel had no ability to distinguish whether the milk or tea was poured first, and decided to test her claim. An experiment was designed in which Muriel would be presented with some cups of tea with the milk poured first, and some cups with the tea poured first. (a) In plain English (no symbols), describe the null and alternative hypotheses for this scenario. (b) Let \(p\) be the true proportion of times Muriel can guess correctly. State the null and alternative hypothesis in terms of \(p\).
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